We consider the following stationary FitzHugh-Nagumo system: \( \left\{ \begin{aligned}&-\varepsilon ^2\Delta U =f(U)- V, \qquad&\text {in}\ \Omega ,\\&-\Delta V+\gamma V =\delta _\varepsilon U,&\text{ in }\ \Omega ,\\&U = V =0,&\text {on}\ \partial \Omega , \end{aligned} \right. \) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^2\) , and \(\varepsilon \) , \(\gamma \) , and \(\delta _{\varepsilon }\) are positive parameters. The nonlinear term f(U) is given by \(U(U-a)(1-U)\) , where a(x) is a smooth positive function in \(C^2(\Omega )\cap C^1(\overline{\Omega })\) with values in \((0,\frac{1}{2})\) . Using the Lyapunov–Schmidt reduction method, we rigorously establish the existence of multi-peaked solutions. Each peak is shown to concentrate near a nondegenerate critical point of the inhomogeneity a(x). Furthermore, we analyze the spectrum of the associated linearized operator. The large eigenvalues(of order O(1)) analysis shows that the constructed multi-peaked solution is linearly unstable. For the small eigenvalues(of order o(1)), we prove that they are of order \(O(\varepsilon ^2)\) , and their rescaled accumulation points are characterized by a finite-dimensional eigenvalue problem involving the Hessian matrices of \(a(x)\) at the concentration points.
